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Spherical splitting of 3-orbifolds

Published online by Cambridge University Press:  10 April 2007

CARLO PETRONIO
CARLO PETRONIO*
Affiliation:
Dipartimento di Matematica Applicata, Università di Pisa, Via Filippo Buonarroti, 1G, I-56127, PISA, Italy. e-mail: [email protected]
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Abstract

The famous Haken–Kneser–Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds would be sufficient to establish it. We demonstrate in this paper that this is not the case, proving that the apparently natural notion of “essential” system of spherical 2-orbifolds is not adequate in this context. We also show that the statement itself of the theorem must be given in a substantially different way. We then prove the theorem in full detail, using a certain notion of “efficient splitting system.”

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 269 - 287
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1] Boileau, M., Leeb, B. and Porti, J.. Geometrization of 3-dimensional orbifolds, math.GT/0010184.Google Scholar
[2] Boileau, M., Maillot, S. and Porti, J.. Three-dimensional orbifolds and their geometric structures. Panor. Synthès. 15 (2003).Google Scholar
[3] Bonahon, F. and Siebenmann, L.. The characteristic toric splitting of irreducible compact 3-orbifolds. Math. Ann. 278 (1987), 441–479.CrossRefGoogle Scholar
[4] Hempel, J.. 3-Manifolds. Ann. of Math. Stud. 86 (1976).Google Scholar
[5] Cooper, D., Hodgson, C. and Kerkhoff, S.. Three-dimensional orbifolds and cone-manifolds. MSJ Mem. 5 (2000).Google Scholar
[6] Martelli, B.. Complexity of 3-manifolds, math.GT/0405250.Google Scholar
[7] Matveev, S. V.. Complexity theory of three-dimensional manifolds. Acta Appl. Math. 19 (1990), 101–130.CrossRefGoogle Scholar
[8] Matveev, S. V.. Algorithmic topology and classification of 3-manifolds. Algorithms Comput. Math. 9 (2003).CrossRefGoogle Scholar
[9] Perelman, G.. Ricci flow with surgery on three-manifolds, math.DG/0303109.Google Scholar
[10] Petronio, C.. Complexity of 3-orbifolds. To appear in Topology Appl.Google Scholar
[11] Takeuchi, Y. and Yokoyama, M.. The realization of the decompositions of the 3-orbifold groups along the spherical 2-orbifold groups. Topology Appl. 124 (2002), 103–127.CrossRefGoogle Scholar
[12] Thurston, W. P.. The geometry and topology of 3-manifolds. Mimeographed notes (Princeton, 1979).Google Scholar